1.

The point(s) on the parabola `y^2 = 4x` which are closest to the circle`x^2 + y^2 - 24y + 128 = 0` is/are

Answer» equation of circle
`x^2+y^2-24y+128=0`
its center=(0,12)
equation of parabola
`y^2=4x`
where a=1
point P`(at^2,2at)=(t^2,2t)`
normal at point (t)
`y=-tx+at^3+2at`
a=1
`y=-tx+t^3+2t`
this equation should be normal for circle as well
(0,12) should be satisfied
`12=t^3+2t`
3+2t-12=0
(t-2)is a factor
after division
`(t-2)(t^2+2t+6)=0`
but `(t^2+2t+6)!=0`
so, (t-2)=0
t=2
putting this value in point P
P(4,4)


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